Optimal control of drug delivery to brain tumors for a test of PDE driven models using the Galerkin finite element method

The Galerkin finite element method is used to examine the a simple polar test case of optimal drug delivery to brain tumors. The PDE driven mathematical model is a system of three coupled reaction diffusion equations involving the tumor cells, the normal tissue and the drugh concentration. An optimal control problem is formualted keeping in mind the primary goals of the treatment, i.e., minimizing the tumor cell density and reducing the side effects of drugs. A distributed parameter method based on application of variational calculus to a pseudo-Hamiltonian, is used to obtain a coupled system of forward state equations and backward co-state equations. The Galerkin form of the finite element method is used due to its greater facility in numerically representing complex structures such as those in the brain. Finally, a two-dimensional circular disk test case is considered and partitioned into a set of rectangular finite elements in polar coordinates with bilinear basis functions on each element, except that triangular elements are used to accommodate the singular origin. Results show significant reduction of the tumor density over time.